A statistical framework to analyze time persistent events is provided
by Donsker-Varadhan large deviation theory. Large deviation theory
deals in general with the exponential decay of probabilities of fluctuations
in random systems, providing an extension of the law of large numbers
and central limit theorem [1]}. Typically one obtains
a large deviation scaling as asymptotic behavior of probability distributions
depending on a small parameter. Donsker-Varadhan large deviation theory
is a particular case of large deviation theory, which deals with the
scaling of the statistics of time averages over a time \(T\) . It predicts
the asymptotic behavior of the probability distribution function of
the time average of an observable for any large enough \(T\)  (where
the small parameter is given by \(1/T\) ), described by a large deviation
rate function. Establishing a large deviation result for a climatic
observable would mean that the probability of extremely rare fluctuations
of the time average of that observable can be inferred from the probability
of much less rare events. For example, the probability of having a
(very rare) heat wave characterized by a value \(a\)  of the temperature
anomaly over several months could be obtained from the probability
of having a (much less rare) heat wave characterized by the same value
\(a\)  of the temperature anomaly over just a few weeks.
